This paper applies homotopy type theory to formal semantics of natural languages and proposes a new model for the linguistic phenomenon of copredication. Copredication refers to sentences where two predicates which assume different requirements for their arguments are asserted for one single entity, e.g., "the lunch was delicious but took forever". This paper is particularly concerned with copredication sentences with quantification, i.e., cases where the two predicates impose distinct criteria of quantification and individuation, e.g., "Fred picked up and mastered three books." In our solution developed in homotopy type theory and using the rule of existential closure following Heim analysis of indefinites, common nouns are modeled as identifications of their aspects using HoTT identity types, e.g., the common noun book is modeled as identifications of its physical and informational aspects. The previous treatments of copredication in systems of semantics which are based on simple type theory and dependent type theories make the correct predictions but at the expense of ad hoc extensions (e.g., partial functions, dot types and coercive subtyping). The model proposed here, also predicts the correct results but using a conceptually simpler foundation and no ad hoc extensions. The proofs in the proposal have been formalized using Agda. Copredication in homotopy type theory (PDF Download Available). Available from: https://www.researchgate.net/publication/320704539_Copredication_in_homotopy_type_theory [accessed Oct 30 2017].

1. Copredication in Homotopy Type
Theory
Hamidreza Bahramian1
Indiana University Bloomington
Presentation based on a joint work with Narges
Nematollahi2 and Amr Sabry3
1 Department of Computer Science
2 Department of Linguistics
3 Department of Computer Science

2. Introduction
Copredication:
the phenomenon where two or more predicates
with different requirements on their arguments
are applied to a single argument
a) The lunch was delicious but took forever.
(Asher, 2011)
b) The heavy book is easy to understand.
(Gotham, 2014)
c) John picked up and mastered three books.
(Asher 2011)

3. History
— Asher(2011), nouns as “dot objects”(Pustejovsky,1995)
◦ i.e., objects that can be viewed under different “aspects”
— Cooper (2011)
◦ nouns as functions from “records” to “record types”
— Luo (2012), Chatzikyriakidisand Luo (2012, 2013, 2015)
◦ common nouns as types
◦ dot types + coercive subtyping
— Gotham(2014)
◦ `book' denotes the set of composite objects physical+informational
◦ criteria of individuation are combined during semantic composition

4. Montague semantics

5. Montague semantics

6. Montague semantics

7. In Luo’s framework

8. In Luo’s framework

9. In Luo’s framework

10. In Luo’s framework

11. In Luo’s framework

12. In Luo’s framework

13. In Luo’s framework

14. In Luo’s framework

15. In Luo’s framework

16. The problem

17. The problem

18. Homotopy Type Theory
— intensional version of Martin-Löf’s type theory [ML75]
— a proof-relevant interpretation of equality
— propositions-as-types principle
— a full cumulative hierarchy of universes
— judgmental vs propositional equality
◦ “=” vs “ ”≡

19. Homotopy Type Theory

20. Homotopy Theory

21. Homotopy Theory

22. Homotopy Theory

23. Homotopy Theory

24. Homotopy Theory

25. Homotopy Theory
Johnson,Chris. " Path Homotopy Animation."Youtube.Youtube, 11 May 2009.Web.
9 July 2016.

26. Homotopy Theory

27. Homotopy Theory
Rowland,Todd. "Homotopic." From MathWorld--AWolframWeb Resource,created by
Eric W.Weisstein. http://mathworld.wolfram.com/Homotopic.html

28. Homotopy Theory

29. Homotopy Theory

30. CN as identifications

31. CN as identifications

32. CN as identifications

33. CN as identifications
Sameness

34. CN as identifications
Sameness Unlikeness

35. CN as identifications

36. CN as identifications
TreePT TreePT =TreePT

37. CN as identifications
TreePT =TreePT
TreePT TreePT

38. CN as identifications
TreePT =TreePT
TreePT TreePT

39. CN as identifications

40. CN as identifications

41. CN as identifications

42. CN as identifications

43. CN as identifications

44. Heim and Kratzer’s framework

45. Heim and Kratzer’s framework

46. Heim and Kratzer’s framework

47. Heim and Kratzer’s framework

48. Heim and Kratzer’s framework

49. Heim and Kratzer’s framework

50. Heim and Kratzer’s framework

51. Heim and Kratzer’s framework

52. Heim and Kratzer’s framework

53. Heim and Kratzer’s framework

54. Semantic computation using HoTT
logic

55. Future work
— Formal semantics of natural languages
◦ “possible world”?
— HoTT
◦ subtyping?
— Homotopy
◦ Prefiguration as theorem?

56. References